Identifier
-
Mp00008:
Binary trees
—to complete tree⟶
Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤ (values match St000442The maximal area to the right of an up step of a Dyck path.)
Values
[.,.] => [[],[]] => [1,0,1,0] => 1
[.,[.,.]] => [[],[[],[]]] => [1,0,1,1,0,1,0,0] => 2
[[.,.],.] => [[[],[]],[]] => [1,1,0,1,0,0,1,0] => 2
[.,[.,[.,.]]] => [[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[.,[[.,.],.]] => [[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
[[.,.],[.,.]] => [[[],[]],[[],[]]] => [1,1,0,1,0,0,1,1,0,1,0,0] => 2
[[.,[.,.]],.] => [[[],[[],[]]],[]] => [1,1,0,1,1,0,1,0,0,0,1,0] => 3
[[[.,.],.],.] => [[[[],[]],[]],[]] => [1,1,1,0,1,0,0,1,0,0,1,0] => 3
[.,[.,[.,[.,.]]]] => [[],[[],[[],[[],[]]]]] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0] => 4
[.,[.,[[.,.],.]]] => [[],[[],[[[],[]],[]]]] => [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0] => 4
[.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0] => 3
[.,[[.,[.,.]],.]] => [[],[[[],[[],[]]],[]]] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4
[.,[[[.,.],.],.]] => [[],[[[[],[]],[]],[]]] => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0] => 4
[[.,.],[.,[.,.]]] => [[[],[]],[[],[[],[]]]] => [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0] => 3
[[.,.],[[.,.],.]] => [[[],[]],[[[],[]],[]]] => [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0] => 3
[[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0] => 3
[[[.,.],.],[.,.]] => [[[[],[]],[]],[[],[]]] => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0] => 3
[[.,[.,[.,.]]],.] => [[[],[[],[[],[]]]],[]] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0] => 4
[[.,[[.,.],.]],.] => [[[],[[[],[]],[]]],[]] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0] => 4
[[[.,.],[.,.]],.] => [[[[],[]],[[],[]]],[]] => [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0] => 3
[[[.,[.,.]],.],.] => [[[[],[[],[]]],[]],[]] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0] => 4
[[[[.,.],.],.],.] => [[[[[],[]],[]],[]],[]] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0] => 4
[.,[.,[.,[.,[.,.]]]]] => [[],[[],[[],[[],[[],[]]]]]] => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => 5
[.,[.,[.,[[.,.],.]]]] => [[],[[],[[],[[[],[]],[]]]]] => [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => 5
[.,[.,[[.,.],[.,.]]]] => [[],[[],[[[],[]],[[],[]]]]] => [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0] => 4
[.,[.,[[.,[.,.]],.]]] => [[],[[],[[[],[[],[]]],[]]]] => [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0] => 5
[.,[.,[[[.,.],.],.]]] => [[],[[],[[[[],[]],[]],[]]]] => [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => 5
[.,[[.,.],[.,[.,.]]]] => [[],[[[],[]],[[],[[],[]]]]] => [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0] => 4
[.,[[.,.],[[.,.],.]]] => [[],[[[],[]],[[[],[]],[]]]] => [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0] => 4
[.,[[.,[.,.]],[.,.]]] => [[],[[[],[[],[]]],[[],[]]]] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0] => 4
[.,[[[.,.],.],[.,.]]] => [[],[[[[],[]],[]],[[],[]]]] => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0] => 4
[.,[[.,[.,[.,.]]],.]] => [[],[[[],[[],[[],[]]]],[]]] => [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0] => 5
[.,[[.,[[.,.],.]],.]] => [[],[[[],[[[],[]],[]]],[]]] => [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => 5
[.,[[[.,.],[.,.]],.]] => [[],[[[[],[]],[[],[]]],[]]] => [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0] => 4
[.,[[[.,[.,.]],.],.]] => [[],[[[[],[[],[]]],[]],[]]] => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0] => 5
[.,[[[[.,.],.],.],.]] => [[],[[[[[],[]],[]],[]],[]]] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0] => 5
[[.,.],[.,[.,[.,.]]]] => [[[],[]],[[],[[],[[],[]]]]] => [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0] => 4
[[.,.],[.,[[.,.],.]]] => [[[],[]],[[],[[[],[]],[]]]] => [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0] => 4
[[.,.],[[.,.],[.,.]]] => [[[],[]],[[[],[]],[[],[]]]] => [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0] => 3
[[.,.],[[.,[.,.]],.]] => [[[],[]],[[[],[[],[]]],[]]] => [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4
[[.,.],[[[.,.],.],.]] => [[[],[]],[[[[],[]],[]],[]]] => [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0] => 4
[[.,[.,.]],[.,[.,.]]] => [[[],[[],[]]],[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0] => 3
[[.,[.,.]],[[.,.],.]] => [[[],[[],[]]],[[[],[]],[]]] => [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0] => 3
[[[.,.],.],[.,[.,.]]] => [[[[],[]],[]],[[],[[],[]]]] => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0] => 3
[[[.,.],.],[[.,.],.]] => [[[[],[]],[]],[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0] => 3
[[.,[.,[.,.]]],[.,.]] => [[[],[[],[[],[]]]],[[],[]]] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0] => 4
[[.,[[.,.],.]],[.,.]] => [[[],[[[],[]],[]]],[[],[]]] => [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0] => 4
[[[.,.],[.,.]],[.,.]] => [[[[],[]],[[],[]]],[[],[]]] => [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0] => 3
[[[.,[.,.]],.],[.,.]] => [[[[],[[],[]]],[]],[[],[]]] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0] => 4
[[[[.,.],.],.],[.,.]] => [[[[[],[]],[]],[]],[[],[]]] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0] => 4
[[.,[.,[.,[.,.]]]],.] => [[[],[[],[[],[[],[]]]]],[]] => [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0] => 5
[[.,[.,[[.,.],.]]],.] => [[[],[[],[[[],[]],[]]]],[]] => [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0] => 5
[[.,[[.,.],[.,.]]],.] => [[[],[[[],[]],[[],[]]]],[]] => [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0] => 4
[[.,[[.,[.,.]],.]],.] => [[[],[[[],[[],[]]],[]]],[]] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0] => 5
[[.,[[[.,.],.],.]],.] => [[[],[[[[],[]],[]],[]]],[]] => [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0] => 5
[[[.,.],[.,[.,.]]],.] => [[[[],[]],[[],[[],[]]]],[]] => [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0] => 4
[[[.,.],[[.,.],.]],.] => [[[[],[]],[[[],[]],[]]],[]] => [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0] => 4
[[[.,[.,.]],[.,.]],.] => [[[[],[[],[]]],[[],[]]],[]] => [1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0] => 4
[[[[.,.],.],[.,.]],.] => [[[[[],[]],[]],[[],[]]],[]] => [1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0] => 4
[[[.,[.,[.,.]]],.],.] => [[[[],[[],[[],[]]]],[]],[]] => [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0] => 5
[[[.,[[.,.],.]],.],.] => [[[[],[[[],[]],[]]],[]],[]] => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0] => 5
[[[[.,.],[.,.]],.],.] => [[[[[],[]],[[],[]]],[]],[]] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0] => 4
[[[[.,[.,.]],.],.],.] => [[[[[],[[],[]]],[]],[]],[]] => [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0] => 5
[[[[[.,.],.],.],.],.] => [[[[[[],[]],[]],[]],[]],[]] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0] => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => [[],[[],[[],[[],[[],[[],[]]]]]]] => [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0] => 6
[.,[.,[.,[.,[[.,.],.]]]]] => [[],[[],[[],[[],[[[],[]],[]]]]]] => [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0] => 6
[.,[.,[.,[[.,.],[.,.]]]]] => [[],[[],[[],[[[],[]],[[],[]]]]]] => [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0] => 5
[.,[.,[.,[[.,[.,.]],.]]]] => [[],[[],[[],[[[],[[],[]]],[]]]]] => [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0] => 6
[.,[.,[.,[[[.,.],.],.]]]] => [[],[[],[[],[[[[],[]],[]],[]]]]] => [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0] => 6
[.,[.,[[.,.],[.,[.,.]]]]] => [[],[[],[[[],[]],[[],[[],[]]]]]] => [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0] => 5
[.,[.,[[.,.],[[.,.],.]]]] => [[],[[],[[[],[]],[[[],[]],[]]]]] => [1,0,1,1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,0] => 5
[.,[.,[[.,[.,.]],[.,.]]]] => [[],[[],[[[],[[],[]]],[[],[]]]]] => [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0] => 5
[.,[.,[[[.,.],.],[.,.]]]] => [[],[[],[[[[],[]],[]],[[],[]]]]] => [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,0] => 5
[.,[.,[[.,[.,[.,.]]],.]]] => [[],[[],[[[],[[],[[],[]]]],[]]]] => [1,0,1,1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0] => 6
[.,[.,[[.,[[.,.],.]],.]]] => [[],[[],[[[],[[[],[]],[]]],[]]]] => [1,0,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,0] => 6
[.,[.,[[[.,.],[.,.]],.]]] => [[],[[],[[[[],[]],[[],[]]],[]]]] => [1,0,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,0] => 5
[.,[.,[[[.,[.,.]],.],.]]] => [[],[[],[[[[],[[],[]]],[]],[]]]] => [1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,0] => 6
[.,[.,[[[[.,.],.],.],.]]] => [[],[[],[[[[[],[]],[]],[]],[]]]] => [1,0,1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0] => 6
[.,[[.,.],[.,[.,[.,.]]]]] => [[],[[[],[]],[[],[[],[[],[]]]]]] => [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0] => 5
[.,[[.,.],[.,[[.,.],.]]]] => [[],[[[],[]],[[],[[[],[]],[]]]]] => [1,0,1,1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => 5
[.,[[.,.],[[.,.],[.,.]]]] => [[],[[[],[]],[[[],[]],[[],[]]]]] => [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0] => 4
[.,[[.,.],[[.,[.,.]],.]]] => [[],[[[],[]],[[[],[[],[]]],[]]]] => [1,0,1,1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0] => 5
[.,[[.,.],[[[.,.],.],.]]] => [[],[[[],[]],[[[[],[]],[]],[]]]] => [1,0,1,1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0] => 5
[.,[[.,[.,.]],[.,[.,.]]]] => [[],[[[],[[],[]]],[[],[[],[]]]]] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0,0] => 4
[.,[[.,[.,.]],[[.,.],.]]] => [[],[[[],[[],[]]],[[[],[]],[]]]] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0,0] => 4
[.,[[[.,.],.],[.,[.,.]]]] => [[],[[[[],[]],[]],[[],[[],[]]]]] => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0] => 4
[.,[[[.,.],.],[[.,.],.]]] => [[],[[[[],[]],[]],[[[],[]],[]]]] => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0] => 4
[.,[[.,[.,[.,.]]],[.,.]]] => [[],[[[],[[],[[],[]]]],[[],[]]]] => [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0,0] => 5
[.,[[.,[[.,.],.]],[.,.]]] => [[],[[[],[[[],[]],[]]],[[],[]]]] => [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0,0] => 5
[.,[[[.,.],[.,.]],[.,.]]] => [[],[[[[],[]],[[],[]]],[[],[]]]] => [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0] => 4
[.,[[[.,[.,.]],.],[.,.]]] => [[],[[[[],[[],[]]],[]],[[],[]]]] => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0,0] => 5
[.,[[[[.,.],.],.],[.,.]]] => [[],[[[[[],[]],[]],[]],[[],[]]]] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0,0] => 5
[.,[[.,[.,[.,[.,.]]]],.]] => [[],[[[],[[],[[],[[],[]]]]],[]]] => [1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0] => 6
[.,[[.,[.,[[.,.],.]]],.]] => [[],[[[],[[],[[[],[]],[]]]],[]]] => [1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0] => 6
[.,[[.,[[.,.],[.,.]]],.]] => [[],[[[],[[[],[]],[[],[]]]],[]]] => [1,0,1,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,0] => 5
[.,[[.,[[.,[.,.]],.]],.]] => [[],[[[],[[[],[[],[]]],[]]],[]]] => [1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0] => 6
[.,[[.,[[[.,.],.],.]],.]] => [[],[[[],[[[[],[]],[]],[]]],[]]] => [1,0,1,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0] => 6
[.,[[[.,.],[.,[.,.]]],.]] => [[],[[[[],[]],[[],[[],[]]]],[]]] => [1,0,1,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0] => 5
[.,[[[.,.],[[.,.],.]],.]] => [[],[[[[],[]],[[[],[]],[]]],[]]] => [1,0,1,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => 5
[.,[[[.,[.,.]],[.,.]],.]] => [[],[[[[],[[],[]]],[[],[]]],[]]] => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0,0] => 5
[.,[[[[.,.],.],[.,.]],.]] => [[],[[[[[],[]],[]],[[],[]]],[]]] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0,0] => 5
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Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
Map
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
This sends the maximal height of the Dyck path to the depth of the tree.
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